Question 1

A decision-maker is faced with a choice between a lottery with a 30% chance of a payoff of $30 and a 70% chance of a payoff of $80, and a guaranteed payoff of $65. If the decision maker's utility function is U = $\sqrt { I }$ , what is the risk premium associated with this choice?

Accepted Answer

First, we need to find the expected utility of each option:

Expected utility of the lottery:
U(0.3*$30 + 0.7*$80) = U($62) = √62 ≈ 7.87

Expected utility of the guaranteed payoff:
U($65) = √65 ≈ 8.06

So, the expected utility of the guaranteed payoff is higher than that of the lottery.

The risk premium is the amount of money that a decision maker is willing to pay to avoid the risk. It can be calculated by finding the difference between the expected payoffs of the two options and the minimum amount of money that would make the decision maker indifferent between the two options. In this case, the minimum amount of money is the risk premium.

Expected payoff of the lottery: 0.3*$30 + 0.7*$80 = $62

Risk premium = Expected payoff of the guaranteed option - Expected payoff of the lottery
= $65 - $62
= $3

However, we need to convert this monetary value into a utility value in order to compare it to the expected utilities. Using the utility function, we can find that the utility of $3 is:

U($3) = √3 ≈ 1.73

Since the expected utility of the guaranteed option is higher than that of the lottery, the decision maker would choose the guaranteed option. Therefore, the best choice is the guaranteed option with a payoff of $65.

Question 2

A decision maker has a utility function $U = \sqrt { I }$ . This decision maker is

Accepted Answer

This utility function exhibits decreasing marginal utility of income, as the square root function increases at a decreasing rate. This means that the decision maker is risk-averse, as they value each additional dollar of income less and less, and thus prefer a certain outcome over a risky one with the same expected value.

Question 3

A risk premium, RP, can be computed with the following formula, where I₁ and I₂ are the two payoffs to a lottery, with probabilities p and (1-p), respectively :

Accepted Answer

The risk premium is the difference between the expected value of a lottery and the certainty equivalent. The certainty equivalent is the amount of money that a person would be willing to accept for sure in lieu of the lottery. The expected value of a lottery is the average amount of money that a person would expect to win if they played the lottery many times. The risk premium is the amount of money that a person is willing to pay to avoid the risk of losing money in a lottery.

Question 4

Suppose a decision maker has a utility function $U = \sqrt { I }$ and is faced with a lottery where there is a 30% chance of earning $30 and a 70% chance of earning $80. What is the expected utility of this lottery?

Accepted Answer

To find the expected utility, we need to calculate the weighted average of the utility of each outcome, where the weights are the probabilities of those outcomes. Thus,
Expected Utility = 0.3 * sqrt(30) + 0.7 * sqrt(80)
Expected Utility = 3.11 + 3.69
Expected Utility = 6.8
Therefore, the expected utility of this lottery is 6.8, which is closest to answer choice B (7.9).

To determine the best choice, we need to compare the expected utility of the lottery to the utility of each alternative. Let's assume there are two options:

Option 1: a guaranteed payment of $50 
Utility of Option 1 = sqrt(50) = 7.07

Option 2: the lottery described above 
Expected Utility of Option 2 = 6.8

Since the utility of Option 1 is greater than the expected utility of Option 2, the decision maker should choose Option 1 (taking the guaranteed payment of $50).

Question 5

Your current disposable income is $10,000. There is a 10% chance you will get in a serious car accident, incurring damage of $1,900. (There is a 90% chance that nothing will happen.) Your utility function is $U = \sqrt { I }$ , where I is income. If this policy is priced at $40, what is the change in your expected utility if you purchase the policy rather than no insurance?

Accepted Answer

The answer of Your current disposable income is $10,000. There...

Question 6

A decision maker has a utility function $U = I ^ { 2 } + 500$ . This decision maker is

Accepted Answer

The utility function $U = I^2 + 500$ is convex, indicating that the decision maker derives increasing marginal utility from wealth, which is characteristic of a risk-loving individual.

Question 7

A decision maker can be described with utility that is only a function of income. If this function is linear, the decision maker is

Accepted Answer

If the utility function is linear, it means that the decision maker's level of satisfaction or happiness is directly proportional to their income level. This type of decision maker is considered risk-neutral because they do not have a preference for risk or uncertainty. They will make decisions based solely on the expected value of the outcomes without considering the variance or probability of those outcomes.

Question 8

A decision maker has a utility function U = 10I. This decision maker is

Accepted Answer

Since the utility function is a linear function of income (I), the decision maker is risk-neutral. The level of income (I) directly determines the level of utility (U) for the decision maker, meaning that they are indifferent to risk when making choices.

Question 9

A risk premium is

Accepted Answer

A risk premium is the amount of additional compensation required to make a decision-maker indifferent between taking on a risky investment and a risk-free investment. In other words, it is the minimum amount of money that someone would need to receive in order to be willing to take on a risky investment with an uncertain outcome. This is calculated as the difference between the expected value of the risky investment and the expected value of a risk-free investment. Therefore, option C is the best choice as it describes the concept of a risk premium accurately.

Question 10

Which of the following statements is correct for a decision maker facing a choice between a sure thing and a lottery when the sure thing has the expected payoff of the lottery?

Accepted Answer

Risk-averse decision makers require a positive risk premium to bear risk because they are more concerned about potential losses than potential gains. They will only take on risk if they expect to be compensated for it with a higher expected payoff than the sure thing. Risk-loving decision makers, on the other hand, are willing to bear risk without a positive risk premium because they are more focused on the potential gains rather than potential losses. Risk-neutral decision makers do not have a preference between risk and certainty and will only require a risk premium if the expected payoff of the lottery is lower than the expected payoff of the sure thing. The characteristics of the lottery can affect the size of the risk premium, but ultimately it is the individual decision maker's utility function that determines their preference for risk. Risk-averse decision makers are those who prefer a certain payoff over the uncertain outcome of a lottery with the same expected payoff. Therefore, they would require a positive risk premium to bear risk. Risk-loving and risk-neutral decision makers may not require a risk premium, or may even be willing to pay a premium for the opportunity to take on risk. The risk premium may depend on the characteristics of the lottery, but it is ultimately determined by the decision maker's preference for risk.

Question 11

A fairly-priced insurance policy is one in which

Accepted Answer

The answer of A fairly-priced insurance policy is one in...

Question 12

Your current disposable income is $10,000. There is a 10% chance you will get in a serious car accident, incurring damage of $1,900. (There is a 90% chance that nothing will happen.) Your utility function is $U = \sqrt { I }$ , where I is income. What is the most you would be willing to pay for this policy (rather than no insurance)?

Accepted Answer

The expected utility without insurance is 0.9 * sqrt(10000) + 0.1 * sqrt(8100) = 99.1. To find the maximum premium, set the utility with insurance equal to this: sqrt(10000 - P) = 99.1. Solving gives P = $199.

Question 13

An insurance company that sells fairly-priced insurance policies to a large number of individuals with similar realized accident risk probabilities should expect to

Accepted Answer

When an insurance company sells fairly-priced insurance policies to a large number of individuals with similar realized accident risk probabilities, it is essentially spreading the risk across many policyholders. The premiums collected should, on average, cover the claims paid out, leading the company to break even, assuming efficient management and accurate risk assessment.

Question 14

Lotteries A and B have the same expected value, but B has larger variance. Which of the following statements is true, all else equal?

Accepted Answer

Risk-averse decision makers prefer less risky options, so they require a higher risk premium for riskier options. Since lottery B has a larger variance, it is considered riskier. Therefore, if the decision-maker is risk-averse, they will require a larger risk premium for lottery B compared to lottery A.

Question 15

A decision-maker is faced with a choice between a lottery with a 30% chance of a payoff of $30 and a 70% chance of a payoff of $80, and a guaranteed payoff of $65. If the decision maker's utility function is $U = I + 500$ , what is the risk premium associated with this choice?

Accepted Answer

To find the risk premium, we need to compare the expected utility of the gamble and the expected utility of the certain payoff.

Expected utility of the gamble: 
0.3(500+30) + 0.7(500+80) = 387

Expected utility of the certain payoff:
500+65 = 565

Since the expected utility of the certain payoff is higher than the expected utility of the gamble, there is no risk premium. Therefore, the decision-maker should choose the certain payoff of $65.

Question 16

Would you expect an insurance company in the "real world" to sell an insurance policy for exactly the "fairly-priced" level as defined in the text?

Accepted Answer

Insurance companies have to make a profit to cover their operational costs and investments. Therefore, they would not sell an insurance policy at exactly the "fairly-priced" level, as there would be no expected value of profits for the insurance company. This is because the expected value of profits for the insurance company would be zero, and the insurance company would not be able to sustain itself in the long run. Insurance companies are in the business of making a profit, and therefore selling a policy at the "fairly-priced" level, which includes all expected costs, would not generate any profit for the company. Therefore, it is unlikely that an insurance company in the "real world" would sell a policy at this level.

Question 17

Consider a fairly-priced insurance policy that fully indemnifies the purchaser against their loss. This insurance policy would most likely be purchased by

Accepted Answer

A risk-averse decision maker would be most likely to purchase this insurance policy because they are more concerned with avoiding a loss than potentially gaining from taking risks. A risk-neutral decision maker may not see the need to fully indemnify against loss because they are indifferent to risk, while a risk-loving decision maker may see purchasing the insurance policy as limiting their potential gains. Therefore, option C is the best choice.

Question 18

Your current disposable income is $10,000. There is a 10% chance you will get in a serious car accident, incurring damage of $1,900. (There is a 90% chance that nothing will happen.) Your utility function is $U = \sqrt { I }$ , where I is income. What is the fair price of this policy?

Accepted Answer

The fair price of the insurance policy is the expected loss, which is 10% of $1,900, equaling $190.

Question 19

Consider an insurance policy with $15,000 worth of coverage. If there is a 10% chance the owner of the policy will file a claim for the $15,000 (and a 90% chance they will not file a claim), a fair price for this policy is

Accepted Answer

The expected value of the policy (the amount the insurance company can expect to pay out on average) is:
(0.10 × $15,000) + (0.90 × $0) = $1,500
Therefore, a fair price for the policy would be $1,500. Option B is the only choice that matches this value.

Question 20

A decision maker can be described with utility which is only a function of income and which exhibits diminishing marginal utility of income. This decision maker is

Accepted Answer

The fact that the decision maker's utility exhibits diminishing marginal utility of income suggests that they value each additional unit of income less than the previous unit. This indicates risk aversion, as they would be less likely to take on risky choices with uncertain outcomes that could potentially result in a loss of income.

Quiz 15: Risk and Information

A decision-maker is faced with a choice between a lottery with a 30% chance of a payoff of $30 and a 70% chance of a payoff of $80, and a guaranteed payoff of $65. If the decision maker's utility function is U = $\sqrt { I }$ , what is the risk premium associated with this choice?

A decision maker has a utility function $U = \sqrt { I }$ . This decision maker is

A risk premium, RP, can be computed with the following formula, where I₁ and I₂ are the two payoffs to a lottery, with probabilities p and (1-p) , respectively :

Suppose a decision maker has a utility function $U = \sqrt { I }$ and is faced with a lottery where there is a 30% chance of earning $30 and a 70% chance of earning $80. What is the expected utility of this lottery?

A decision maker has a utility function $U = I ^ { 2 } + 500$ . This decision maker is

A decision maker can be described with utility that is only a function of income. If this function is linear, the decision maker is

A decision maker has a utility function U = 10I. This decision maker is

A risk premium is

Which of the following statements is correct for a decision maker facing a choice between a sure thing and a lottery when the sure thing has the expected payoff of the lottery?

A fairly-priced insurance policy is one in which

An insurance company that sells fairly-priced insurance policies to a large number of individuals with similar realized accident risk probabilities should expect to

Lotteries A and B have the same expected value, but B has larger variance. Which of the following statements is true, all else equal?

A decision-maker is faced with a choice between a lottery with a 30% chance of a payoff of $30 and a 70% chance of a payoff of $80, and a guaranteed payoff of $65. If the decision maker's utility function is $U = I + 500$ , what is the risk premium associated with this choice?

Would you expect an insurance company in the "real world" to sell an insurance policy for exactly the "fairly-priced" level as defined in the text?

Consider a fairly-priced insurance policy that fully indemnifies the purchaser against their loss. This insurance policy would most likely be purchased by

Your current disposable income is $10,000. There is a 10% chance you will get in a serious car accident, incurring damage of $1,900. (There is a 90% chance that nothing will happen.) Your utility function is $U = \sqrt { I }$ , where I is income. What is the fair price of this policy?

Consider an insurance policy with $15,000 worth of coverage. If there is a 10% chance the owner of the policy will file a claim for the $15,000 (and a 90% chance they will not file a claim) , a fair price for this policy is

A decision maker can be described with utility which is only a function of income and which exhibits diminishing marginal utility of income. This decision maker is

Quiz 15: Risk and Information

A decision-maker is faced with a choice between a lottery with a 30% chance of a payoff of $30 and a 70% chance of a payoff of $80, and a guaranteed payoff of $65. If the decision maker's utility function is U = I\sqrt { I }I​ , what is the risk premium associated with this choice?

A decision maker has a utility function U=IU = \sqrt { I }U=I​ . This decision maker is

A risk premium, RP, can be computed with the following formula, where I1 and I2 are the two payoffs to a lottery, with probabilities p and (1-p) , respectively :

Suppose a decision maker has a utility function U=IU = \sqrt { I }U=I​ and is faced with a lottery where there is a 30% chance of earning $30 and a 70% chance of earning $80. What is the expected utility of this lottery?

A decision maker has a utility function U=I2+500U = I ^ { 2 } + 500U=I2+500 . This decision maker is

A decision maker can be described with utility that is only a function of income. If this function is linear, the decision maker is

A decision maker has a utility function U = 10I. This decision maker is

A risk premium is

Which of the following statements is correct for a decision maker facing a choice between a sure thing and a lottery when the sure thing has the expected payoff of the lottery?

A fairly-priced insurance policy is one in which

An insurance company that sells fairly-priced insurance policies to a large number of individuals with similar realized accident risk probabilities should expect to

Lotteries A and B have the same expected value, but B has larger variance. Which of the following statements is true, all else equal?

A decision-maker is faced with a choice between a lottery with a 30% chance of a payoff of $30 and a 70% chance of a payoff of $80, and a guaranteed payoff of $65. If the decision maker's utility function is U=I+500U = I + 500U=I+500 , what is the risk premium associated with this choice?

Would you expect an insurance company in the "real world" to sell an insurance policy for exactly the "fairly-priced" level as defined in the text?

Consider a fairly-priced insurance policy that fully indemnifies the purchaser against their loss. This insurance policy would most likely be purchased by

Your current disposable income is $10,000. There is a 10% chance you will get in a serious car accident, incurring damage of $1,900. (There is a 90% chance that nothing will happen.) Your utility function is U=IU = \sqrt { I }U=I​ , where I is income. What is the fair price of this policy?

Consider an insurance policy with $15,000 worth of coverage. If there is a 10% chance the owner of the policy will file a claim for the $15,000 (and a 90% chance they will not file a claim) , a fair price for this policy is

A decision maker can be described with utility which is only a function of income and which exhibits diminishing marginal utility of income. This decision maker is

A decision-maker is faced with a choice between a lottery with a 30% chance of a payoff of $30 and a 70% chance of a payoff of $80, and a guaranteed payoff of $65. If the decision maker's utility function is U = $\sqrt { I }$ , what is the risk premium associated with this choice?

A decision maker has a utility function $U = \sqrt { I }$ . This decision maker is

A risk premium, RP, can be computed with the following formula, where I₁ and I₂ are the two payoffs to a lottery, with probabilities p and (1-p) , respectively :

Suppose a decision maker has a utility function $U = \sqrt { I }$ and is faced with a lottery where there is a 30% chance of earning $30 and a 70% chance of earning $80. What is the expected utility of this lottery?

A decision maker has a utility function $U = I ^ { 2 } + 500$ . This decision maker is

A decision-maker is faced with a choice between a lottery with a 30% chance of a payoff of $30 and a 70% chance of a payoff of $80, and a guaranteed payoff of $65. If the decision maker's utility function is $U = I + 500$ , what is the risk premium associated with this choice?

Your current disposable income is $10,000. There is a 10% chance you will get in a serious car accident, incurring damage of $1,900. (There is a 90% chance that nothing will happen.) Your utility function is $U = \sqrt { I }$ , where I is income. What is the fair price of this policy?